This is my current DNS example.com.db zone file. I followed a tutorial. It works, because when I point to this DNS from another server via resolve.conf, it will actually forward me to the right IP when I do "ping example.com".

Incrementing the serial is only important if other DNS servers are fetching DNS from his server. If it's a purely local server with only DNS clients using it, the serial number isn't such a big deal.
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SchofNov 10 '09 at 0:56

If I understand your question correctly, you're very close to where you want to go:

Assuming you want dbserver1.example.com, and not dbserver1.otherdomain.com, do the following:

dbserver1 IN A 44.245.66.222
cacheserver1 IN A 38.221.44.555

If you want to go to some other domain (as your comment indicates) then you'd want:

dbserver1.otherdomain.com. IN A 44.245.66.222
cacheserver1.otherdomain.com. IN A 38.221.44.555

(Note the terminating period on the domain names above. Very important; otherwise you'll get dbserver1.otherdomain.com.example.com.)

Moving away from the specific topic of BIND config files to the general topic of DNS, I'm not sure what you're trying to do, but it doesn't sound like a good idea.

The Internet moved away from hosts files to DNS because maintaining a bunch of hosts files quickly became impossibly arduous. If you're using DNS to create your own hosts file, I'd suggest:

Using an actual hosts file.

Not doing it at all and using DNS as DNS.

If you just want to be able to go to dbserver1.example.com by typing "ping dbserver1" then you need to follow Yves' suggestion and add "search example.com" to your /etc/resolv.conf. This is completely independent of your DNS server -- it's a local setting on each machine telling it to append "example.com" to invalid addresses.